Fourier analysis methods in operator ergodic theory onsuper-reflexive Banach spaces

Earl Berkson

doi:10.3934/era.2010.17.90

Article Contents

2010, Volume 17: 90-103. Doi: 10.3934/era.2010.17.90

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Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces

Earl Berkson^1,

1.
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801

Received: June 2010

Published: January 2010

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Abstract

On reflexive spaces trigonometrically well-bounded operators (abbreviated "twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates "transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(\mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(\mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.

Keywords:

Ergodic Hilbert transform,
super-reflexive Banach space,
spectral decomposition,
p-variation,
trigonometrically well-bounded operator.

Mathematics Subject Classification: Primary: 26A45, 46B20, 47A35, 47B40.

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